The easy way is to have lots of Ethers, Turbo Ethers, and a Elixir if you have it. Inflict Deshell, Imperil, and Poison on him. I think slow works on Ereshkigal too. Have one garb that helps a lot with guarding and give it your best , preferably quiet guardian or cyber jumpsuit. Equip the item that Snow drops on the defense garb. Now, when the fight starts, quickly overclock and inflict Deshell, Imperil, Slow if it works, and Poison on him. Then sit and watch as his HP slowly drops. It's long and boring, but the easiest way to beat it. Repeat the process over and over, using Ethers and healing when you need to, and guarding his attacks when he goes to attack you.
If you want to do an offensive way, I guess, equip a powerful magic garb with all of your most powerful magical weapons and whatnot, and give it a fully powered Elementa, Element, Deshell, and Imperil. Spira's Summoner (DLC garb), comes with a fully powered Elementa so it's useful for the entire game. Use the same defense set up in the above paragraph I wrote, and have lots of healing items. As soon as the fight begins, overclock, and use Deshell and Imperil until it's inflicted on him, then spam Elementa or Element, whichever you like to stagger him. When overclock runs out, quickly do it again and then Elementa/Element until he staggers. You have to stagger him four times to actually get some serious damage, I believe. So just keep doing this until you have him staggered 4 times and then just keep overclocking, healing, restoring EP, and Ereshkigal should go down sooner or later. I hope this helps!